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HBO is removed from the surface into the bulk gas by diffusion through a boundary layer.

The driving force for diffusion is the concentration difference between the surface and the bulk gas, which in Equation (2.47) is converted to partial pressure through the ideal gas law (Equation (2.9)).

The boundary layer thickness and the mass transfer coefficient relies on fluid dynamics, like for diffusion in the melt. The fluid dynamic mass transfer coefficient for diffusion of HBO (kHBO) and the boundary layer thickness may be calculated from the dimensionless Sher-wood number (Sh) in Equation (2.48). Reynolds number (Re) in Equation (2.49) and the Schmidt number (Sc) in Equation (2.50) are the other dimensionless numbers for transport of mass.Re relates to the flow characteristics and its value indicate whether the flow is laminar (relatively lowRe) or turbulent (high Re).

Sh =kHBOL

In these equations,ρ is the density and μ is the dynamic viscosity of the fluid. L is a charac-teristic length related to the geometry andv is the bulk flow velocity. Correlations between the dimensionless numbers are fitted to experiments or derived from models for different of geometries and flow regimes.

A schematic of the impinging jet flow pattern from a nozzle that is perpendicular to the surface is shown in Figure 2.7. For a circular nozzle, the characteristic length is the diameter of the nozzle exit (L = d), and studies of impinging jets refer to the flow at this position. Re through the nozzle exit is calculated by Equation (2.51), and relates directly to the flow rate (Q) through the mean flow velocity ¯v = _πd^4Q2.

The flow is axisymmetric around the middle of the nozzle, and is stagnant in the stagnation point at the surface below. Figure 2.7 also identifies the stagnation region (0 ≤ r ≤ rs) in which the flow is deflected by the impingement surface into the wall jet region (rs≤ r ≤ rw), and it definesz as vertical distance and H as the height of the nozzle exit. Entrainment and shear from the ambient gas also start to slow down the free jet at the edges and cause it to broaden away from the nozzle. The gas slows down significantly in the wall jet region as it spreads over the surface and drags gas from the ambient. Computational fluid dynamic (CFD) modeling of a laminar impinging jet onto molten silicon by Næss [55] predicts the radial gas velocity along the surface to scale linearly with the gas velocity at the nozzle exit.

Like in most studies, the following theory is presented for a flat surface. The melt surface is however typically not flat. For instance, a dimple can form in the stagnation region [11], and induction can shape the surface as a dome. Fluid dynamic calculations by Nordstrand and Tangstad [17] did not indicate a dimple in their experiments, which are similar to experiments in this work, but ripples on the surface that is expected to increase the reaction area.

Figure 2.7: Schematic of impinging jet flow pattern (modified from Bergman et al. [72]).

Most studies consider uniform impinging jets, in which turbulence through a nozzle produce a uniform flow velocity at the nozzle exit (shown in Figure 2.7), while a tubular lance used by Nordstrand and Tangstad [17] for reactive gas refining has a nonuniform, parabolic velocity profile [73]. Re was also in the laminar region (Re < 5000 [74]). Scholtz and Trass [75]

developed correlations forSh for such a setup in the wall jet region in 1963 [74] and in the stagnation region in 1970 [75]. They found the mass transfer coefficient in the stagnation region to be higher for parabolic velocity profile compared to uniform flow by a factor of 2.2 at the stagnation point and 1.6 beneath the edge of the nozzle (^r_d = 0.5). In Equations (2.52)-(2.53), expressions for Sh and Re in Equations (2.48) and (2.51) are inserted into their original equations to show the dependence of the mass transfer coefficient on gas flow rate and nozzle diameter separately. The conditions these correlations are verified within are given in Table 2.6.

Scholtz and Trass [75] found mass transfer to be independent of nozzle height for 0.25 ≤

H

d ≤ 6 with nozzle diameters between 1.9 cm ≤ d ≤ 5.2 cm. In this range, the highest mass transfer coefficients are observed at the stagnation point (Equation (2.52)), and it decreases monotonically away from this point at r = 0. The radial variation is however not large within the stagnation region as shown in Figure 2.8, and Equation (2.52) agreed with

exper-Table 2.6: Conditions for Equations (2.52)-(2.53) and (2.56). *Assumptions in text.

iments at least for ^r_d < 0.1. Consistently, Altenberend [18] found the boron removal rate in plasma refining experiments to vary insignificantly with nozzle heights between 30− 80 mm (0.74 ≤ ^H_d ≤ 1.98). For^H_d < 0.25, mass transfer rates are increased away from the stagnation point to a maximum near the nozzle edge. Differences between jet impingement on a melt and on a solid plate can be expected, particularly for such low nozzle heights, as the gas jet impingement can make a dimple in the surface or penetrate into the liquid.

Scholtz and Trass [75] identified the stagnation region within 0 ≤ ^r_d ≤ 0.6 and presented an approximate correlation for mass transfer of which Equation (2.52) is a special case. The correlation agreed with experiments with naphthalene-air (Sc = 2.45) within the entire ex-perimental range of 500≤ Re ≤ 1960 and 0.05 ≤ ^H_d ≤ 6 as seen for the curve “Theoretical - stagnation region” in Figure 2.8.

In the wall jet region, Equation (2.53) [74] agrees with experimental data and the exact so-lution of momentum, continuity and mass conservation equations under the conditions given in Table 2.6. The radius in which Equation (2.53) applies decrease below _d^r < 25 for de-creasingRe < 1000 due to boundary layer separation. This was accompanied by vortex formation, which resulted in stagnant fluid and abrupt decrease in the mass transfer coeffi-cient forRe < 500. The fluid dynamic mass transfer coefficient can be obtained over both stagnation and wall jet regions for lowRe (375). Scholtz and Trass [75] found Equation (2.53) to intersect with the correlation for the stagnation region at^r_d = 0.6 as shown in Figure 2.8. This consistency was not obtained for higherRe (1970).

Figure 2.8: Mass transfer in stagnation and wall jet regions for Re = 375, Sc = 2.45, 0.25 ≤ ^H_d ≤ 6 (modified from Scholtz and Trass [75]). R = ¹₂d is the radius of the nozzle exit.

It is seen from Equation (2.52) that the mass transfer coefficient increases with gas flow rate asks,HBO∝ Q¹²in the stagnation region directly beneath the lance. This is found for the whole stagnation region. In the wall jet region,kw,HBO∝ Q³⁴and the mass transfer coefficient decreases with radial distance askw,HBO∝ r⁻⁵⁴. The mass transfer coefficient decreases with nozzle diameter askHBO∝ d⁻¹at the stagnation point andkHBO∝ d⁻¹²in the wall jet region due to decreased flow velocity.

The mass transfer coefficient also depends on the viscosity of the gas. The effect of viscosity for mass transfer to the gas can be studied by comparing removal rates in atmospheres with different inert gases. Boron removal do not appear sensitive to the atmosphere gas as in gas blowing experiments with steam in argon and nitrogen by Safarian et al. [41], for which they reported total mass transfer coefficients ofkt= 7 μm/s and kt= 6 μm/s, respectively.

Nakamura et al. [11] added helium to argon as plasma gas, but no effect on boron removal was observed. Furthermore, they did not observe an increasing flow rate to directly affect boron removal. Consequently, mass transfer to the gas was not considered to be rate determining.

Finally, diffusion coefficient of gases increases with increasing temperature and decreases (inversely proportional) with total pressure, but is almost independent of composition for a given binary gas pair [66]. The temperature dependence of diffusion rates can however be masked by chemical equilibrium, which typically scales exponentially with temperature as an Arrhenius relation.

Since the mass transfer coefficient and thus the flux is not constant over the reaction area, it must be averaged over the crucible cross-section area (a flat surface is assumed) in order to relate it to the removal rate according to Equation (2.54). Due to the axisymmetric ge-ometry of the impinging jet flow, cylindrical coordinates are used for integration over the crucible cross-section areaAc= _2π

0

_rc

0 rdrdΘ, where rcis the radius of the crucible and Θ is the angle of integration. The average mass transfer coefficient also explicitly shows the dependence on crucible radius. Integration of the mass transfer coefficient in Equation (2.55) is possible ifkHBOin the stag-nation region is assumed constant to Equation (2.52) and that it intersects with (2.53) for the wall jet region atrs≈ 0.6d for Re = 300 like for Re = 375 in Figure 2.8, so that the integral is valid for ^r_d ≤ 15 like Equation (2.53). The radial integration limit is the lower of the cru-cible radius (rc) and the radius of the wall jet region (rw) used in Equation (2.56). Equation (2.56) shows the dependence of the average mass transfer coefficient on the ratio of nozzle diameter to crucible radius and the gas flow rate.

¯kHBO= 1

The wall jet was found dominating for the average mass transfer in impinging jet experiments by Næss [55], which can also be expected for typical setups with crucible cross-section areas orders of magnitude larger than the stagnation area. Næss [55] found JSi ∝ Q³⁴ for oxi-dation of silicon in a setup similar to that of Nordstrand and Tangstad [17], although at a slightly larger scale. Thus, a dependence close to ¯kHBO ∝ Q³⁴ can be expected. Equation (2.56) reduces to Equation (2.57) when the contribution of the stagnation region is assumed negligible, and this equation is valid for a larger range ofSc as shown in Table 2.6.

¯kHBO≈ 0.5082DHBOSc¹³

The nozzle diameter has two contributions to mass transfer in the wall jet region. The de-creased gas velocity through a wider lance decreases ¯kHBO ∝ d⁻¹². A thicker lance also increases the area that is not included in the wall jet (the stagnation area), which gives an offset from proportionality as the second term of the fraction in Equation (2.57) decreases decreases proportionally to−d¹⁴.

If the crucible radius is longer than the wall jet region (rw> rcin Equation (2.56)), the max-imum area of the impinging jet is utilized. The total rate of boron removal is then expected to be independent of crucible radius, and the average mass transfer coefficient decreases as k¯HBO ∝ r_c⁻². If the crucible radius is shorter than the wall jet radius, the crucible radius becomes the upper integration limit for the wall jet region andrw= rc. The dependence on crucible radius in the first term in the fraction of the wall jet integral becomes ^r_r^3/4c2

c = rc−⁵₄. The effect of a second stagnation region along the crucible wall, which deflects the wall jet if rw> rc, is not accounted for by Equations (2.56) and (2.57).

Boron removal through step 3 proceeds at its maximum rate when the bulk concentration of HBO equals zero in Equation (2.47). If all other steps are fast in comparison, there is equilibrium at the interface with the bulk content of boron in the melt. Insertion of Equation (2.38) into (2.47) with pi,H2 ≈ p_b,H2 for fast supply of hydrogen to the surface provides Equation (2.58).

The mass transfer coefficient in Equation (2.58) shows the same dependence on reactant concentrations as the equilibrium distribution of boron at the interface (Equation (2.4)). The partial pressure of SiO at the interface can not be simplified to the bulk concentration of steam because steam reacts with SiO above the surface as explained in Section 2.3.6, and only a fraction of the supplied steam reaches the surface.

Furthermore, the SiO partial pressure at the interface is expected to show a radial dependence for impinging jets, because it found that the partial pressure of SiO on the surface is deter-mined by the flux of oxygen to the surface [55]. The partial pressure of SiO on the surface can thus be expected to decrease outwards in the wall jet region, not only because of its fluid dynamic mass transfer coefficient, but also the fact that the bulk content of steam decreases as it is consumed while it flows over the surface in the wall jet region. This suggests the flux of boron to decrease faster with radial distance compared tokHBO. Additionally, the as-sumption of negligible partial pressure of HBO in the bulk may not hold in the entire wall jet region even if step 3 is rate determining, due to accumulation of HBO in the bulk gas flow. An expression for the rate of boron removal if step 3 is rate determining (integration of Equation (2.58)) is not complete without a model for active oxidation by steam, which is

not fully developed (Section 2.3.6), and fluid dynamic modeling for accumulation of HBO (Section 2.3.5).